Uniform-grid spatial bin index¶

A flat regular-grid index that retrieves local spatial candidates for exact relation or distance checks.

Version: v1 · 2026-10-17 · History
Domain-specific #: 14346
Domain group: Applied Sciences & Engineering
Origin domain: Computer Science & Software Engineering
Subdomains: Computational Geometry, Spatial Data Structures → Computer Science & Software Engineering
Aliases: Spatial bin grid, Bin (computational geometry)

Core Idea¶

A uniform-grid spatial bin index overlays a spatial domain with regular cells, records which items occupy those cells, and retrieves nearby candidate items by looking in cells relevant to a query. The grid is a candidate filter: sharing a bin or appearing in a searched stencil does not by itself establish intersection or neighbor status. An exact relation or distance test makes that final decision.^[1]^[2]

This is the reusable mechanism behind the selected computational-geometry “bin” seed. It is not a frequency bin in a histogram. A geometric segment may be registered in every cell it crosses, while an atom in a molecular simulation may be registered in one cell and compared with atoms from a neighborhood of cells. Those unlike carriers preserve coordinate-based enrollment, local retrieval, and subsequent exact checking.^[1]^[2]

Structural Signature¶

Regular coordinate cells. A flat grid divides the relevant space into cells that can be addressed from location. Without regular cells, the mechanism becomes another kind of spatial index.^[1]^[2]
A location ownership convention. Every coordinate location has one owning cell, including locations on grid lines under a declared boundary rule. This partitions locations, even though one extended item may intersect several cells. The boundary rule is a correctness convention to state for a particular implementation; it is not a claim that both inspected sources prescribe the same rule.
Item enrollment. References to items are retained in the cells appropriate to their geometry. A segment can traverse several cells; a point atom occupies one bin in the LAMMPS construction. Multi-cell occupancy is therefore a variant, not a universal rule.^[1]^[2]
Local candidate retrieval. A geometric operation compares elements sharing a cell, or a proximity search scans a stencil of nearby cells. If every item is scanned regardless of location, the bin index is not doing its filtering work.^[1]^[2]
Declared cell resolution. Cell size determines how many bins must be visited and how many items may crowd each bin. No one width is optimal for every object distribution or query.^[1]^[2]
Handoff to an exact predicate. An intersection or distance test checks retrieved candidates when an exact answer is required. That test is downstream of the index; co-bin membership alone is only a possible-match signal.^[1]^[2]

The first five roles define the index. The final role defines how its candidate output is used without mistakenly making a particular intersection or force computation part of the index's identity.

What It Is Not¶

It is not histogram binning, which accumulates counts by value range rather than retaining spatial item references for local candidate generation. It is not a quadtree or binary space partition, whose cells arise through recursive or adaptive subdivision. Those structures may solve related retrieval problems, but a flat regular grid is the differentia of this entry.^[1]^[2]

It is also not a complete exact-intersection or force-evaluation algorithm. Franklin's co-cell segment pairs still require intersection checking; LAMMPS scans candidate atoms and then tests whether their separation is below the neighbor-list cutoff. Treating every candidate as a confirmed match would overcount or introduce false positives.^[1]^[2]

Scope of Application¶

The directly inspected cases are geometric intersection detection and molecular-dynamics neighbor-list construction. Franklin describes an overlaid uniform grid whose cells collect traversing geometric elements; LAMMPS describes a regular two- or three-dimensional bin grid and a nearby-bin stencil for atom pairs. Both use the same spatial filter while differing in carrier, enrollment, and search neighborhood.^[1]^[2]

Other spatial queries may use a similar grid, but their exact predicate and coverage rule must be specified. The sources here do not establish that every implementation supports rectangle queries, incremental insertions and deletions, linked lists, or a bin width equal to the physical force cutoff.^[1]^[2]

Clarity¶

For a proposed bin index, state the cell geometry and width, how boundary coordinates are assigned, how each item is enrolled in cells, which cells a query visits, and what exact condition is checked afterward. These answers expose whether the method can miss a qualifying pair or report the same pair more than once. Duplicate handling is a property of a particular search or list convention, not a constitutive bin role.^[1]^[2]

In LAMMPS the neighbor-list cutoff is the force cutoff plus a skin distance, and the documentation describes a bin width around half that neighbor-list cutoff for many typical cases. Saying merely “one bin per force cutoff” would confuse three different quantities: the physical interaction range, the list's buffer range, and grid resolution.^[2]

Manages Complexity¶

Spatial location first narrows a potentially large item set to cells that could contain relevant candidates. This allows the exact and potentially expensive relation test to operate on a smaller local set. Franklin's author account explains the grid as a way to cull object pairs that may intersect; LAMMPS uses bins and a stencil to build neighbor lists instead of checking every atom pair in its normal bin method.^[1]^[2]

The simplification is conditional. An unfortunate grid resolution or distribution can create crowded cells or too much bin traversal; Franklin warns of poor worst-case behavior. LAMMPS documents smaller-bin stencil overhead and larger-bin extra distance calculations, and provides a quadratic no-bin fallback for small sparse systems. The index does not promise an all-input complexity bound.^[1]^[2]

Abstract Reasoning¶

The decisive counterfactual is to remove the grid while keeping the same exact predicate. The answer to an intersection or distance query should remain the same, but candidate generation would lose the locality filter and may require many more comparisons. If removing the exact predicate instead changes confirmed results into mere candidates, the index and the decision test have been correctly separated.^[1]^[2]

For correctness, reason from coverage: every genuinely qualifying pair must appear together in a searched cell or stencil. For efficiency, reason from occupancy and visits: how many cells are searched and how many candidates survive to an exact test? The two questions must not be conflated; a grid can be correct yet inefficient for a particular workload.

Knowledge Transfer¶

The mechanism transfers between line segments on a geometric screen and atom positions in a simulation because both have spatial coordinates, regular cell membership, and local candidate searches. The per-item enrollment and exact predicate do not transfer unchanged: extended segments span cells, while LAMMPS places an atom in a bin and searches neighboring bins before a cutoff-distance test.^[1]^[2]

The name “bin” alone does not transfer the mechanism to categorical counts, temporal buckets, or arbitrary database partitions. A literal instance needs spatial coordinates, regular cells, and cell-limited candidate retrieval.

Examples¶

Canonical: geometric segment intersection¶

Franklin describes overlaying a one-level uniform grid, finding the cells traversed by each geometric element, and comparing pairs that share a cell. For line-segment intersection, the shared-cell relation supplies candidates; the segment-intersection predicate supplies confirmed crossings. Mapped back: a flat grid supplies addressable cells; segments are enrolled in traversed cells; shared-cell comparison retrieves local candidates; a declared cell resolution controls crowding; exact intersection remains downstream. The directly inspected author account does not specify how the grid resolution was selected for this case.^[1]

Applied: LAMMPS atom neighbor-list build¶

LAMMPS overlays the simulation domain with regular neighbor bins. Each atom is assigned to a bin, and a precomputed stencil selects nearby bins for each central atom. A pair is added only after its distance is checked against the neighbor-list cutoff. Mapped back: the simulation grid supplies cells; atom indices supply enrollment; the stencil supplies local retrieval; configurable bin size governs lookup and check costs; the distance comparison is the downstream predicate. LAMMPS rebuilds its neighbor lists from scratch as needed, so incremental deletion is not part of this example.^[2]

Structural Tensions¶

T1: Finer cells vs. fewer cell visits. Finer bins can reduce unrelated items per cell and the number of exact candidate checks, while increasing stencil traversal or repeated enrollment for extended objects. These costs cannot both be minimized by shrinking bins without limit. Diagnostic: At the current width, is work dominated by visiting bins or by checking candidates? LAMMPS documents this bin-size tradeoff, and Franklin warns that uniform-grid performance depends on the input.^[1]^[2]

Structural–Framed Character¶

The entry is mostly structural within spatial computation. Vocabulary travel: “bin” travels widely, but the literal test here requires coordinate cells and candidate retrieval. Evaluative weight: the grid is a method, not a promise of superior speed; workload determines performance. Institutional origin: no institution makes a bin index valid; its enrollment and coverage rule do. Human-practice dependence: a designer chooses resolution, but operation depends on geometry and data access rather than a social convention. Import versus recognition: segments and atoms literally instantiate the same local-index relation; a histogram borrows the word while changing the role of cells. Its character: mixed-structural, because the grid-and-retrieval relation transfers across unlike spatial carriers while its literal correctness still depends on coordinates and geometric coverage.^[1]^[2]

Structural Core vs. Domain Accent¶

The liftable constituent is the Partition Prime's division of a carrier set into mutually exclusive, collectively exhaustive blocks. Here the carrier is the coordinate domain, and regular cells are its blocks; a boundary convention gives each location one owner. Index is a related key-to-location side-table pattern, but its full auxiliary-structure and update commitments are not required by every flat spatial bin. Remove spatial coordinates, regular cells, item enrollment, and geometric coverage, and the result is a broader partition or retrieval pattern rather than this domain-specific index. A cross-domain analogy to organizing documents in folders lacks those spatial correctness conditions and is not an instance of this entry.

This entry is part of Partition.

The catalog's Partition Prime is an internal constituent: regular grid cells divide coordinate locations into one owning cell apiece, enabling cell addresses and local lookup. That is distinct from item enrollment, which may span cells. Spatial Indexing is a close neighbor, but its live full signature requires typical output-sensitive retrieval cost; a correct flat grid with crowded bins or many false candidates need not meet it. The live domain-specific Quadtree and Binary Space Partitioning entries are nearby alternatives whose recursive subdivisions do not make them parents of this flat-grid identity.

Relationships to Other Abstractions¶

Current abstraction Uniform-grid spatial bin index Domain-specific

Parents (1) — more general patterns this builds on

Uniform-grid spatial bin index is part of Partition Prime

The grid's mutually exclusive, collectively exhaustive coordinate cells are an internal partition that makes cell addressing and local retrieval possible.

Hierarchy path (1) — routes to 1 parentless root

Uniform-grid spatial bin index → Partition → Set and Membership

Neighborhood in Abstraction Space¶

Uniform-grid spatial bin index sits in a sparse region of the domain-specific corpus (94^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2721 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-17

Not to Be Confused With¶

Frequency bins. A histogram stores counts over value intervals; this index stores spatial item references for local lookup.
A confirmed collision list. Co-bin membership generates candidates; an exact predicate confirms the desired relation.^[1]^[2]
Universal multi-cell occupancy. Geometric segments can cross many cells; an atom in the inspected LAMMPS build occupies one bin.^[1]^[2]
Automatic dynamic maintenance. The inspected LAMMPS neighbor list is rebuilt from scratch as needed; no universal incremental deletion rule is established.^[2]

References¶

[1] W. Randolph Franklin, “Geometry Applications,” §5, “Uniform Grids”, first-person author account, directly inspected. It describes flat-grid traversal and co-cell pair comparison, with workload-dependent performance. The linked 1989 original paper was identified but its PDF body was not directly opened for this draft; no unverified grid-parameter rule is attributed to it. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u

[2] LAMMPS project, Developer Guide 30Sep2026, §4.4.3, “Neighbor lists”, official release documentation whose displayed git info was 30Sep2026 when inspected; especially paragraphs on neighbor cutoff and rebuilding, regular bins and stencils, atom enrollment, exact distance check, and bin-size tradeoff. The live documentation URL may later advance. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w